Of course, a wiki-answer of "contemporary art" yields a definition of art created in the last half of the 20th century up until our current 21st century era. But what does "contemporary art" really mean? Who coined the term?
One of my professors at Brooklyn College answered a student's question in a way that may help resolve this, at least somewhat. The kid said, "I don't get it. It's a black cube. Like, $100 worth of material. In the Met. That thing in the Met costs five million dollars. Why?"
Short answer: It's in the Met. Long answer: It was a first. Which requires a long answer involving Contemporary Art.
Art is a market. Contemporary means "of the time". But the market wants to be one step ahead. Therefore, "Contemporary Art" is a term not coined, but rather devised. It is, or has been, devised by a market which seeks to draw investment into future holdings, much like stocks. If you invest in that cube, in that formaldehyde-tank-with-shark, in that diamond-encrusted skull, you stand to reap rewards.
The terms "first", "second", and "third-world" were indeed coined by the military. "First" world was Communist block. "Second", the "Free World". "Third", developing countries, i.e., the poor nations ("third world country"). In the "Free World" (Capitalism), art is a commodity and that's why we have contemporary art. It is a competition, a way to one-up the challenger.
The Western system of values is not at all universal. It has affected all civilized countries on Earth, but not necessarily embraced. Contemporary Art is a Western construct and it is Capitalist. That is not meant as a negative remark, simply, objectively, true. And it is American in that the Europe and America of today don't spend much time on tradition. There is rich history of art in Europe, to be sure, and Africa, and China, and everywhere beyond the Western perimeters, but the only tradition from America exists with its native peoples. Therefore, "Contemporary Art" was not "coined", as per your question, but rather, assimilated into the vernacular of our post-20th century times.
On a personal note, one of my favorite artists is Andy Warhol. Hungarian/Slovak by decent, he loved to proclaim himself "American! I'm as American as it gets." So, I really don't want to disdain the system of what "Contemporary Art" is, it is, to use a terrible cliche, what it is. And that's what it is.
I am a professional artist in the Western world. I hope my perspective upon this may be of use. Thank you, J Chase